5 · dynamical systems

families of evolution operators, invariant sets and poincare maps

A continuous-time dynamical system on a manifold $M$ is a family of evolution operators $$ \{\varphi_t\}_{t\in\mathbb{R}} $$ satisfying the group (or semi-group) property $$ \varphi_{t+s}=\varphi_t\circ\varphi_s,\qquad \varphi_0=\mathrm{id}, $$ and depending continuously (or smoothly) on $t$ and on the initial condition. Each orbit $$ \mathcal{O}(x)=\{\varphi_t(x)\mid t\in\mathbb{R}\} $$ is the image of a group homomorphism from $(\mathbb{R},+)$ into $\operatorname{Diff}(M)$. An invariant set $\Lambda\subset M$ satisfies $\varphi_t(\Lambda)=\Lambda$ for all $t$. Compact invariant sets that are isolated (i.e. the maximal invariant set in some neighborhood) are the principal objects of qualitative analysis.

When a periodic orbit $\Gamma$ of period $T$ intersects a local cross-section $\Sigma$ transversally at a point $p$, the first-return (Poincare) map $$ P:\Sigma\supset U\to\Sigma,\qquad x\mapsto\varphi_{\tau(x)}(x) $$ is well-defined in a neighborhood of $p$. The fixed point $p$ of $P$ corresponds to $\Gamma$, and the spectrum of the linearization $DP(p)$ (the Floquet multipliers) determines the stability of the orbit: if all multipliers lie inside the unit circle, $\Gamma$ is asymptotically stable. Transversality of the crossing guarantees that $P$ is a local diffeomorphism, so the local dynamics transverse to $\Gamma$ are completely captured by a discrete map of one lower dimension. This reduction is the principal motivation for the introduction of Poincare maps: questions of stability, bifurcation, and uniqueness of periodic orbits are thereby converted into fixed-point problems for smooth maps.

definition (continuous dynamical system / flow). a $C^r$ flow on a manifold $M$ is a $C^r$ map $\varphi:\mathbb{R}\times M\to M$, written $\varphi_t(x)=\varphi(t,x)$, such that $\varphi_0=\mathrm{id}$ and $\varphi_{t+s}=\varphi_t\circ\varphi_s$ for all $t,s\in\mathbb{R}$ (when both sides are defined, if only a local flow is present). each $\varphi_t$ is a $C^r$ diffeomorphism of $M$ in the global complete case.

definition (orbit and invariant set). the orbit of $x$ is $\mathcal{O}(x)=\{\varphi_t(x):t\in\mathbb{R}\}$. a set $\Lambda\subset M$ is invariant if $\varphi_t(\Lambda)=\Lambda$ for all $t$. it is isolated if there is a compact neighborhood $N$ of $\Lambda$ such that $\Lambda$ is the maximal invariant set contained in $N$.

definition (local cross-section). a local cross-section (or local transversal) to a flow at $p$ is an embedded disc $\Sigma$ through $p$ with $$ T_p M=T_p\Sigma\oplus\mathrm{span}\{v(p)\}, $$ where $v(p)=\frac{d}{dt}|_0\varphi_t(p)$ is the velocity of the flow (assumed nonzero).

theorem (existence of the poincare map). let $\Gamma$ be a periodic orbit of period $T$ through $p$, with $\dot\varphi_t(p)\neq 0$, and let $\Sigma$ be a local cross-section at $p$. then there exist a neighborhood $U\subset\Sigma$ of $p$ and a unique $C^r$ return-time function $\tau:U\to\mathbb{R}$ with $\tau(p)=T$ such that $$ P(x):=\varphi_{\tau(x)}(x)\in\Sigma $$ for all $x\in U$, and $P(p)=p$. moreover $P:U\to\Sigma$ is a $C^r$ local diffeomorphism onto its image.

proof. consider the map $$ F:\mathbb{R}\times\Sigma\to\mathbb{R}^k,\qquad F(t,x)=\pi_\nu\bigl(\varphi_t(x)-p\bigr) $$ in a chart where $\Sigma$ is open in a hyperplane through $p$ with normal projection $\pi_\nu$ along $v(p)$ (more invariantly: a smooth tubular neighborhood chart of $\Sigma$ with transverse coordinate vanishing exactly on $\Sigma$). then $F(T,p)=0$ and $$ \partial_t F(T,p)=\pi_\nu\bigl(v(\varphi_T(p))\bigr)=\pi_\nu(v(p))\neq 0 $$ by transversality. the implicit function theorem yields a unique $C^r$ function $\tau$ near $p$ with $\tau(p)=T$ and $F(\tau(x),x)=0$, i.e. $\varphi_{\tau(x)}(x)\in\Sigma$. set $P(x)=\varphi_{\tau(x)}(x)$. smoothness of $P$ follows from smoothness of the flow and of $\tau$. since $\varphi_t$ is a local diffeomorphism in $x$ and the graph of $\tau$ is transverse, $DP(p)$ is invertible on $T_p\Sigma$ (its kernel would produce a direction in $T_p\Sigma$ mapped into the flow direction only, contradicting the direct sum splitting). by the inverse function theorem $P$ is a local $C^r$ diffeomorphism. $\square$

definition (floquet multipliers). the eigenvalues of $DP(p):T_p\Sigma\to T_p\Sigma$ are the Floquet multipliers of $\Gamma$ (the nontrivial multipliers of the monodromy $D\varphi_T(p)$ transverse to the flow).

theorem (asymptotic stability from the poincare map). if every Floquet multiplier of $\Gamma$ satisfies $|\mu|<1$, then $\Gamma$ is asymptotically stable as a periodic orbit: there is a neighborhood $W$ of $\Gamma$ such that $\mathrm{dist}(\varphi_t(x),\Gamma)\to 0$ as $t\to+\infty$ for every $x\in W$.

proof. hyperbolic contraction of the discrete map $P$ at the fixed point $p$ (all eigenvalues of $DP(p)$ inside the open unit disc) implies, by the discrete Hartman-Grobman theorem or by a standard discrete Lyapunov estimate, that iterates $P^n(x)\to p$ for $x\in\Sigma$ near $p$. any point $y$ near $\Gamma$ hits $\Sigma$ in finite time under the flow (flow-box about the orbit), and subsequent returns follow the iterates of $P$. continuous dependence and compactness of $\Gamma$ convert return convergence into $\mathrm{dist}(\varphi_t(y),\Gamma)\to 0$. $\square$

remark. if some multiplier satisfies $|\mu|>1$, $\Gamma$ is unstable. multipliers on the unit circle require center-manifold or Lyapunov analysis (bifurcation chapter).

suspension of discrete dynamical systems

Conversely, every discrete dynamical system generated by a diffeomorphism $f:M\to M$ may be realized as the Poincare map of a continuous flow. The suspension (or mapping torus) construction proceeds as follows. On the product $M\times[0,1]$ impose the identification $$ (x,1)\sim(f(x),0). $$ The resulting quotient manifold $M_f$ carries the suspension flow $$ \psi_t(x,s)=(x,s+t) $$ (with the obvious wrapping under the identification). The cross-section $M\times\{0\}$ is transverse to $\psi_t$ and the first-return map is precisely $f$. Thus the qualitative theories of flows and of diffeomorphisms are equivalent up to this canonical embedding; every phenomenon observed for maps reappears as a special case of a flow, and vice versa. The construction also supplies a geometric realization of discrete symmetries and of forced oscillations as autonomous flows on an enlarged phase space.

definition (suspension manifold). for a diffeomorphism $f:M\to M$ of class $C^r$, the suspension is the quotient $$ M_f=(M\times[0,1])/\sim,\qquad (x,1)\sim(f(x),0), $$ with the quotient $C^r$ structure descending from the product because $f$ is a $C^r$ diffeomorphism.

theorem (suspension realizes the map as a poincare return). the vector field $\partial/\partial s$ on $M\times[0,1]$ descends to a $C^{r-1}$ (resp. $C^r$ if care is taken at the gluing) nonsingular vector field $X_f$ on $M_f$ whose flow $\psi_t$ satisfies: the section $\Sigma=M\times\{0\}\subset M_f$ is a global cross-section, every orbit returns to $\Sigma$ in time $1$, and the first-return map is $f$.

proof. on the open set $M\times(0,1)$ the field $\partial_s$ is smooth. under the identification $(x,1)\sim(f(x),0)$ the field is preserved because translation in $s$ is compatible with the discrete jump by $f$ at the ends: flowing time $1$ from $(x,0)$ reaches $(x,1)\sim(f(x),0)$. thus $\partial_s$ descends to a continuous (and $C^{r-1}$ if $f\in C^r$) vector field $X_f$ never zero. the flow on representatives is $\psi_t([x,s])=[x,s+t]$ with integer parts of $s+t$ implemented by applying $f$ (or $f^{-1}$) the appropriate number of times. starting at $[x,0]$, the first positive time with $s$-coordinate wrapping back to $0$ is $t=1$, and the return point is $[f(x),0]$. hence the Poincare map on $\Sigma$ is $f$. $\square$

remark. forced time-periodic ODEs $\dot x=v(x,t)$ with $v(\cdot,t+T)=v(\cdot,t)$ suspend as autonomous systems on the extended space $M\times S^1$ with angular speed $2\pi/T$; the Poincare map is the stroboscopic (period) map.

topological stability and global geometry of phase space

A flow $\varphi_t$ is topologically stable if every sufficiently $C^1$-small perturbation $\psi_t$ is topologically equivalent to $\varphi_t$ via a homeomorphism close to the identity. For flows possessing only hyperbolic equilibria and hyperbolic periodic orbits (Morse-Smale systems) topological stability holds on compact manifolds. The proof relies on the persistence of transverse intersections of stable and unstable manifolds and on the structural stability of hyperbolic fixed points of the associated Poincare maps.

When the phase space is non-compact, or when non-hyperbolic behavior is admitted, topological stability may fail. Nevertheless the global geometry remains tightly constrained by the existence of Lyapunov functions, index theory, and the topology of the underlying manifold. Isolated invariant sets possess Conley indices that are invariant under continuation, while the Morse inequalities relate the Betti numbers of the phase space to the number and type of equilibria. In the special case of a gradient flow the global attractor is determined by the critical points of the potential and the connecting orbits between them; the resulting Morse complex recovers the homology of the manifold.

Thus the theory of dynamical systems organizes the phase space into a skeleton of invariant sets (equilibria, periodic orbits, homoclinic and heteroclinic connections) whose local stability is decided by linearization or by Lyapunov functions, whose mutual arrangement is governed by transversality, and whose persistence under perturbation is controlled by hyperbolicity. The Poincare map and the suspension construction ensure that the same geometric picture applies equally to continuous and discrete time, furnishing a unified language for the qualitative analysis of all finite-dimensional evolution equations.

definition (topological equivalence of flows). two flows $\varphi_t$ and $\psi_t$ on $M$ are topologically equivalent if there is a homeomorphism $h:M\to M$ sending oriented orbits of $\varphi$ onto oriented orbits of $\psi$ (reparametrization of time along orbits is allowed).

definition (hyperbolic equilibrium for a flow). an equilibrium $x_*$ is hyperbolic if no eigenvalue of $Dv(x_*)$ has zero real part (chapter 2).

definition (hyperbolic periodic orbit). a periodic orbit is hyperbolic if no Floquet multiplier has modulus $1$ (equivalently: $1$ is a simple monodromy eigenvalue along the flow direction and no other multipliers lie on the unit circle).

definition (morse-smale flow, compact case). a $C^1$ flow on a compact manifold is Morse-Smale if it has finitely many hyperbolic equilibria and hyperbolic periodic orbits, the stable and unstable manifolds of these critical elements intersect transversally, and every orbit has alpha and omega limit sets among these critical elements (the nonwandering set is exactly the critical set).

theorem (structural stability for hyperbolic fixed points of maps). if $p$ is a hyperbolic fixed point of a $C^1$ diffeomorphism $f$ (no eigenvalue of $Df(p)$ of modulus $1$), then for every $C^1$-small perturbation $g$ there is a unique fixed point $p_g$ near $p$, still hyperbolic and of the same index, and the local dynamics of $f$ and $g$ near these fixed points are topologically conjugate.

proof. write $g=f+\eta$ with $\eta$ small in $C^1$. the fixed-point equation $g(x)=x$ rearranges as $(f-\mathrm{id})(x)=-\eta(x)$. hyperbolicity of $f$ at $p$ means that $Df(p)-I$ is invertible on the ambient space when one works in a chart, or more invariantly that $1$ is not in the spectrum of $Df(p)$. the inverse function theorem (or contraction for the map $x\mapsto x-(Df(p)-I)^{-1}(g(x)-x)$) produces a unique $p_g$ near $p$ with $g(p_g)=p_g$, and $Dg(p_g)$ stays hyperbolic for $\eta$ small. local topological conjugacy of hyperbolic fixed points is the discrete Hartman-Grobman theorem: there is a local homeomorphism conjugating $f$ near $p$ to $Df(p)$ near $0$, and similarly for $g$ and $Dg(p_g)$; hyperbolic linear maps with the same index (dimensions of stable/unstable spaces) are topologically conjugate by linear algebra of contracting/expanding splittings. $\square$

theorem (morse inequalities for gradient flows). let $f$ be a smooth Morse function on a compact manifold $M$ (all critical points nondegenerate), and let $c_k$ be the number of critical points of index $k$. if $b_k=\dim H_k(M;\mathbb{R})$ are the Betti numbers, then $$ c_k-c_{k-1}+\cdots+(-1)^k c_0 \ge b_k-b_{k-1}+\cdots+(-1)^k b_0 $$ for each $k$, and $\sum_k (-1)^k c_k=\chi(M)=\sum_k (-1)^k b_k$.

proof (outline via morse complex). the negative gradient flow of $f$ has a hyperbolic equilibrium at each critical point. choosing a generic Riemannian metric, stable and unstable manifolds of critical points intersect transversally (Smale). the Morse chain group $C_k$ is the free abelian group generated by critical points of index $k$; the boundary operator counts signed isolated connecting orbits of relative index $1$. standard gluing and compactness for gradient trajectories imply $\partial^2=0$. the resulting homology is isomorphic to $H_*(M)$ (Morse homology theorem). weak inequalities between the ranks of the chain groups $c_k=\mathrm{rank}\,C_k$ and the Betti numbers $b_k$ are then elementary linear algebra on a chain complex (the Morse inequalities), and the alternating sum identity is the Euler characteristic of both the Morse complex and $M$. $\square$

remark (conley index). for an isolated invariant set $\Lambda$ the Conley index is the homotopy type of a pointed quotient $N/L$ built from an isolating block $(N,L)$; it is invariant under continuation of the flow. it extends Morse data beyond hyperbolic critical points and is deferred to specialized treatments; the existence of such an invariant is the precise sense in which isolated invariant sets leave a topological footprint.

remark (topological stability of morse-smale flows). on a compact manifold, a Morse- Smale flow is structurally stable in the $C^1$ topology: nearby flows are topologically equivalent via a homeomorphism close to the identity. the argument combines persistence of hyperbolic critical elements (preceding theorem and the analogous result for Poincare maps of hyperbolic periodic orbits), persistence of transverse intersections of invariant manifolds, and a filtration argument controlling the global connection graph. full details are classical (Palis-Smale) and lengthy; the geometric moral is that hyperbolicity plus transversality rigidifies the entire orbit skeleton.

summary

Dynamical systems theory organizes the phase space into a skeleton of invariant sets (equilibria, periodic orbits, homoclinic and heteroclinic connections) whose local stability is decided by linearization or by Lyapunov functions, whose mutual arrangement is governed by transversality, and whose persistence under perturbation is controlled by hyperbolicity. The Poincare map and the suspension construction ensure that the same geometric picture applies equally to continuous and discrete time, furnishing a unified language for the qualitative analysis of all finite-dimensional evolution equations.

exercises

exercise 1 (invariant sets). for the linear flow $\varphi_t(x)=e^{tA}x$ on $\mathbb{R}^2$ with $A=\mathrm{diag}(-1,1)$, describe the orbits and list all linear subspaces that are invariant. which of them are isolated as invariant sets?

exercise 2 (poincare section for a cycle). take $\dot r=0$, $\dot\theta=1$ in polar coordinates (on $\mathbb{R}^2\setminus\{0\}$). using the ray $\Sigma=\{\theta=0,\,r>0\}$ as a cross-section, compute the Poincare map $P$ and its derivative at a point of the unit circle. interpret the Floquet multiplier.

exercise 3 (suspension of a rotation). let $f:S^1\to S^1$ be rotation by angle $\alpha$. describe the suspension flow on the torus $S^1\times S^1$ and determine when every orbit is periodic versus dense on a torus leaf.

exercise 4 (morse count on $S^2$). construct a Morse function on $S^2$ with exactly two critical points (height function). check the Morse inequalities and the Euler characteristic against $b_0=1$, $b_1=0$, $b_2=1$, $\chi(S^2)=2$.

exercise 5 (stable manifold intuition on a saddle). for $\dot x=x$, $\dot y=-y$ on $\mathbb{R}^2$, write $W^s(0)$ and $W^u(0)$ explicitly and verify they are the eigenspaces. for the nonlinear saddle $\dot x=x$, $\dot y=-y+\mu x^2$, argue why $W^s$ remains the $y$-axis while $W^u$ bends into the parabola $y=(\mu/3)x^2$ (you may check invariance of that graph).

exercise 6 (omega-limit of a gradient flow). let $\dot x=-\nabla f(x)$ on a compact Riemannian manifold with $f$ Morse. show that every $\omega$-limit set is a critical point. deduce that there are no periodic orbits, using the strict decrease of $f$ away from $\operatorname{Crit}(f)$.